We investigate numerically and analytically a recently proposed model for food webs [Nature 404, 180 (2000)] in the limit of large web sizes and sparse interaction matrices. We obtain analytical expressions for several quantities with ecological interest, in particular, the probability distributions for the number of prey and the number of predators. We find that these distributions have fast-decaying exponential and Gaussian tails, respectively. We also find that our analytical expressions are robust to changes in the details of the model.